San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

On the Equality of the Cross Differences
in the Binding Energies of Nuclei

Let n and p be the number of neutrons and protons, respectively, and BE the binding energy of a nuclide. The incremental binding energy IBEn of a neutron (the first difference in binding energy with respect to the number of neutrons) is

IBEn(n, p) = BE(n, p) − BE(n-1, p)

Likewise the incremental binding energy IBEp of a proton is

IBEp(n, p) = BE(n, p) − BE(n, p-1)

The cross differences of binding energies are then

Δpn²(n, p) = IBEn(n, p) − IBEn(n, p-1)
Δnp²(n, p) = IBEp(n, p) − IBEp(n-1, p)

There is theoretical justification for these to be measuring the binding energy interaction of the last neutron and the last proton to be added to the nuclide. Therefore Δpn²(n, p) should equal Δnp²(n, p). In fact, they are equal by definition as is shown below:

Δpn²(n, p) = BE(n, p) − BE(n-1, p) − BE(n, p-1) + BE(n-1, p-1)
Δnp²(n, p) = BE(n, p) − BE(n, p-1) − BE(n-1, p) + BE(n-1, p-1)

It is also convenient to represent Δpn²(n, p) as the slope of the relationship between IBEn and the number of protons and Δnp²(n, p) as the slope of the relationship between IBEp and the number of neutrons. Below is shown an example of the comparison of such relation ships.

Visually the slopes of the two relationships appear to be approximately equal. The ranges over which the incremental binding energies can be computed are quite different. Here are the relationships for the cases in which n=36 and p=26.

The two quantities nearly coincide over a range from 26 to 35. The limits of that range have to do with the values of 26 and 36.

Visually the two look close but the numerical values deviate. Here are the data upon which the graph above was constructed.

The Incremental Binding
Energies for One Nucleon
as a Function of the Number
of sthe Other Nucleon
Number of
Other Nucleons
19 3 0.1
20 3.2 1.4
21 4 1.51
22 5.1 3.13
23 5.6 2.88
24 7.02 4.153
25 6.9 4.8852
26 8.052 7.3814
27 8.48 7.5291
28 9.658 8.8538
29 9.9102 9.2127
30 11.0599 10.1836
31 11.2264 10.5591
32 12.395 11.9534
33 12.27 12.0453
34 13.71 13.225
35 13.1 13.327
36 16.1 14.479
37 16.2 14.63
38 15.6
39 15.5
40 16.7
41 17.4
42 17.8

The slope of the relationship between IBEn and p over the range of 26 to 35 is 0.56089 MeV per neutron-proton interaction whereas it is 0.66062 MeV per neutron-proton interaction for the relationship between IBEp and n over the same range.

Here is the graph of the data for the slightly different case of p=24. Again there is again the near matching of the slopes of the relationships even though the levels differs more than in the previous case.

Here is an illustration in which the slopes of the relationships for the incremental binding energy of neutrons is noticably different from the those for the incremental binding energy of protons.

The difference in slopes can be explained by the difference in shells involved. The incremental binding energies of neutrons involved the neutron numbers being in the 51 to 82 neutron shell and the proton numbers being in the 29 to 50 proton shell whereas the data for the incremental binding energies of proton involve the proton numbers and the neutron numbers both being in the 29 to 50 shells.

(To be continued.)